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\author{IOnut Danaila
\and Pascal Joly 
\and Sidi Mahmoud Kaber 
\and Marie Postel}
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12.2不可压缩Navier-Stokes方程

不可压缩流体的二维流场完全由速度向量$q = (u(x, y)， v(x, y))\in R^2$，压力$p(x, y)\in R$函数是以下守恒定律的解(例如，

赫希,1988)\cite{2007An}:

•质量守恒:
\[div(q)=0\]
或者，用散度算子的显式表示，
\[
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
\]
•紧化形式的动量守恒方程
\[ \frac{\partial u}{\partial x} +div(q\otimes q)=-Gp+\frac{1}{Re} \Delta q \]
或者，在显式形式中，

\[
\left\{
\begin{aligned}
&  \frac{\partial u}{\partial t}+\frac{\partial u^{2}}{\partial x}+\frac{\partial uv}{\partial y}=
-\frac{\partial p}{\partial x}+\frac{1}{Re}(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}})\\
&   \frac{\partial v}{\partial t}+\frac{\partial uv}{\partial x}+\frac{\partial v^{2}}{\partial y}=
-\frac{\partial p}{\partial y}+\frac{1}{Re}(\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}})
\end{aligned}
\right.
\]


前面的方程以无因次形式写成，使用

以下缩放变量:
\[
x=\frac{x^{*}}{L},y=\frac{y^{*}}{L},u=\frac{u^{*}}{V_0},v=\frac{v^{*}}{V_0},t=\frac{t^{*}}{L/V_0},p=\frac{p^{*}}{\rho_0/V_0^{2}}
\]

其中上标$(*)$表示以物理单位测量的变量的

常数$L,V_0,$分别为表征为模拟流动的参考长度和速度。无量纲数$R-e$称为雷诺数

对惯性(或对流)项的相对重要性进行编号和量化

流动中的粘性(或扩散)3项:

\[Re=\frac{V_0L}{\nu} \]
其中$\nu$是流体的运动粘度。

总而言之，偏微分方程的Navier-Stokes系统将在数值上

在本项目中解决的问题由(12.2)和(12.4)定义;初始条件为

（T = 0)，边界条件将在下面的章节中讨论。

12.4计算域，交错网格，和边界条件
数值求解Navier-Stokes方程是相当简化的

的矩形域Lx × Ly(见图12.1)

到处都是边界条件。速度q(x, y)的周期性

压力$p(x,y)$场在数学上表示为

\[
q(0,y)=q(L_x,y),\qquad p(0,y)=p(L_x,y),\qquad  \forall y \in [0, L_y]\]
\[q(x,0)=q(x,L_y),\qquad p(x,0)=p(x,l_y),\qquad  \forall x \in [0, L_x]
\]

计算解的点分布在

区域遵循矩形和均匀的二维网格。由于在我们的方法中并非所有的变量表都共享相同的网格，因此我们首先定义一个主网格(参见

流体动力学:二维Navier-Stokes方程的求解

图12.1)沿x方向取$n-x$个计算点，沿y方向取$n_y$个计算点生成:
\[
x_c(i)=(i-1)\delta x,\delta x=\frac{L_x}{n_x-1},\qquad i=1\dots n_x
\]
\[
y_c(j)=(j-1)\delta j,\delta j=\frac{L_y}{n_y-1},\qquad i=1\dots n_y
\]

\begin{tikzpicture}
    \draw (0,0) rectangle (1.5,1);
    \draw (1.6,0) rectangle (3.1,1);
    
    [x radius=1,y radius=0.5];
    \end{tikzpicture}
    \\图12.1
\\图12.1。计算域，交错网格和边界条件。

次级网格由初级网格单元的中心定义:

\[
x_m(i)=(i-1/2)\delta x,i=1,\dots n_{xm}
\]
\[
y_m(j)=(j-1/2)\delta y,j=1,\dots n_{ym}
\]

其中我们使用了速记符号$n_{xm}=n_x -1,n_{ym}=n_y -1$内部

定义为矩形$[x_c(i),X_c(i+1)]\times[y_c(j),y_c(j+1)]$的计算单元，未知变量u, v, p将被计算为近似的

不同空间位置的解决方案:

•$u(i, j)\approx u(x_c(i)， ym(j))$ (cell的西面)，

•$v(i, j)\approx v(x_m(i)， y_c(j))$(胞体南侧)，

•$p(i, j)\approx p(x-m(i)， y_m(j))$(细胞中心)。

这种交错排列的变量有很强的优点

压力和速度之间的耦合。它也有帮助(参见参考文献

本章结束)，以避免一些稳定性和收敛的问题，如经验的并配安排(其中所有的变量计算

在相同的网格点)



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